In what follows, we derive closed-form expressions for the critical points of the synchronisation-manifold's stability [Eqs. (REF )-(REF )] in 2 specific cyclic graphs, including their critical link densities, \(\rho _c = k/(N-1)\) . We focus on \(k\) -cycle graphs, \(\mathcal {C}_N(k)\) , and \(k\) -Möbius ladders, \(\mathcal {M}_N(k)\) . \(\mathcal {C}_N(k)\)  are cyclic graphs with even degrees where connections span \(k\)  neighbours per node in ring-like structure (also known as Wiley-Strogatz-Girvan networks [1]}). \(\mathcal {M}_N(k)\)  are our generalisation of the Möbius ladder [2]}, [3]}, which has \(k = 3\) . We introduce \(\mathcal {M}_N(k)\)  graphs to increase the degree to \(3 \le k \le N-1\) , but keeping their overall ladder-like topology. Our derivations for \(\mathcal {C}_N(k)\)  and \(\mathcal {M}_N(k)\)  include finite size critical points and thermodynamic limits.
